Hi OP,
First thing I want to do is give you a big hug! 
I think you are very brave saying all that you have on here that takes a lot of courage. You have said you need help and don't even know where to start. I know how you feel and so do many others so please do not think you are alone.
You sound like you are stuck in a relentless circle and to break it is going to be hard. The house is depressing you even more but your to depressed to do anything about it, your not getting anywhere near enough sleep to cope, if you don't nap you wont be able to function, if you do nap you get nothing done that you want to. You see yourself only in a negative light and validate it to yourself you are useless because if you weren't you would be superwoman and a domesticated goddess. Do you constantly berate yourself in your own head?
So here's my advice to you - (don't know if it will help but here goes)
Be kinder to yourself, speak to yourself the way you would a friend.
If your depression isn't being treated make an appointment with the GP, if it is maybe your meds are needing changed?
Before you do anything like setting yourself all these things you want to change at one go, stop its to much. Do not set yourself up to fail. Set yourself one small change and when you can manage that every day until its water off a ducks back then add something else. Write a list by all means of everything you want to change, but one at a time.
Your toddlers sleep is definitely something that needs to be sorted out for both of you, I haven't got an idea how to do that yet! Surely we can all come up with something between us. Do you have a health visitor you like?
Tackle the lack of sleep then work on all the other things.
Could you do an internet shop save you going and free a day up at weekends?
Put on your ebay listings you only post on certain days many do.
Do you have any friends or other family support?
There is a group run here for toddlers its not much per session and is on twice a week for 2 hours they go on their own. They can go once or twice a week its up to the parent. Is there anything like that in your area?
You haven't been getting out the house much except for the school and work runs do you not like going out on your own? Or is it just lack of energy? Even if you walk your toddler to the first lamp post then home again, day two make it to the second lamp post, its progress and its more than you did before (I had to do this felt so stupid but it worked)
I used to wait hoping I would get up one morning and I would be back to normal, I thought the old me would come back all by herself. My heart goes out to you and please do not pay any attention to any negativity or cutting comments, don't take them to heart, they people don't know you.
Stay strong you can do this, you can xx